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We carry on this process until the specified time is reached. 8: Configuration of block marching technique. 9: Discretization in three directions. y y x 24 Advanced Differential Quadrature Methods In the block-marching technique, the length scales in the x and y directions are fixed as Lx and Ly for different blocks. In this way, we can use the same DQ weighting coefficients in the Lx and Ly directions at different blocks. However, in the time direction, the length scale may be different at different blocks.

16) As n increases, function value becomes larger and larger, and entirely fails to approximate the function f . In other words, polynomial interpolation can be so bad that it does not yield the correct approximation at all. This situation can be avoided if we have freedom to choose the interpolation points for the interval [a, b]. Chebyshev nodes in the following are known to be a good choice. 18) Using Chebyshev nodes, we therefore obtain the following error bounds for polynomial interpolation Approximation and Differential Quadrature f −p ≤2 b−a 4 n max a≤x≤b f (n) (ξn ) n!

It is not necessary for the two virtual points to be δ apart from the boundary point. Then Eq. 41) and Eq. 42) are used to compute the weighting coefficients, exactly the same as in the direct ordinary DQ method. 48). Take an F-C beam as an example to describe the procedures for applying the multi-boundary conditions. Note that the boundary points are 2 and N − 1. 86) Eight methods in applying the multi-boundary conditions are summarized. Although all methods could be used in two-dimensional problems, such as thin plate problems, some limitations exist in some of the methods.

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